Happy Birthday, Emmy Noether

On March 23, 1882, a girl named Emmy Noether was born in Erlangen, Bavaria. The daughter of a mathematician, she would turn out to be a mathematical genius and make one of the most important contributions to physics in the twentieth century. Its impact is only now beginning to be fully appreciated. Noether would be considered one of the foremost feminist heroines of the twentieth century, had more people understood mathematics and physics.

Because she was a woman, Noether was only allowed to audit classes in mathematics at Erlangen University, where her father taught. Still, in 1903 she qualified for the German equivalent of a bachelor's degree. After the university relaxed some of its restrictions against women, in 1907 she completed a dissertation on invariants for ternary biquadratic forms.

For the next eleven years, thanks to the sponsorship of the great mathematician David Hilbert, Noether taught at Erlangen and then Göttingen. However, she received no pay. In 1918 she was given an untenured professorship and by 1923 she was receiving a small salary. Despite many academic accomplishments, however, she was never given tenure, nor was she elected to the Göttingen Academy of Sciences.

Being a Jew, Noether was dismissed from her post upon the Nazi rise to power. In 1933, she emigrated to the U.S. and took a position at Bryn Mawr. Tragically, she died of uterine cancer just two years later, at age 53. Einstein wrote the New York Times that Noether was "the most significant creative mathematical genius thus far produced since the higher education of women began."

Not being a mathematician, I cannot testify to Emmy Noether's mathematical accomplishments. As a physicist, however, I can tell you that the theorem she published in 1915, which was not taught in any class I took on my way to a Ph.D. in physics, completely changed my understanding of the nature of physical law.

I always thought, as most physicists still think, that the laws of physics are restrictions on the behavior of matter that are somehow built into the structure of the universe. Although she did not put it in these terms, Noether derived a theorem that implied otherwise.

Noether's theorem proves that for every continuous space-time symmetry there exists a conservation principle. Three conservation principles form the foundational laws of physics: conservation of energy, conservation of linear momentum, and conservation of angular momentum.

Noether showed that conservation of energy follows from time translation symmetry; conservation of linear momentum follows from space translation symmetry; and conservation of angular momentum follows from space rotation symmetry.

What this means in practice is that when a physicist makes a model that does not depend on any particular time, that is, one designed to work whether it is today, yesterday, 13 billion years ago, or 13 billion years in the future, that model automatically contains conservation of energy. The physicist has no choice in the matter. If she tried to put violation of energy conservation into the model, it would be logically inconsistent.

If another physicist makes a model that does not depend on any particular place in space, that is, one designed to work whether it is in Oxford, Timbuktu, on Pluto, or on the recently discovered galaxy MACS0647-JD that is 13.3 billion light years away, that model automatically contains conservation of linear momentum. The physicist, once again, has no choice in the matter. If he tried to put violation of linear momentum conservation into the model, it would be logically inconsistent.

And, any model that is designed to work with an arbitrary orientation of a system must necessarily contain conservation of angular momentum.

Since these three principles form the basis of classical mechanics, it can be said that the laws of physics do not govern the behavior of matter. They govern the behavior of physicists.

Although this is not widely recognized, Noether's connection between symmetries and laws can be applied beyond space-time to the abstract internal space occupied by the state vector of quantum mechanics. These abstract spaces are not so obvious as the three dimensional space we experience. They are introduced to describe observed forces and transformations. By noting what particular quantities are conserved we can infer what symmetries apply.

In this case, we have a principle called gauge symmetry, which is equivalent to rotational symmetry in multidimensional state vector space. The various conservation principles that apply in this regime, in particular conservation of electric charge, arise from gauge symmetry. Furthermore, when the equation of motion of a charged particle is made locally gauge symmetric, that is, independently symmetric at every point in space and time, Maxwell's equations of electromagnetism fall right out of the mathematics. In other words, the electric and magnetic fields are fictitious forces, like the centrifugal and Coriolis forces, inserted into theories to preserve certain symmetries.

Gauge symmetry did not stop with classical physics. In the late 1940s it was applied to quantum electrodynamics and in the 1970s to the highly successful standard model of elementary particles that received its final corroboration last year with the observation of the Higgs boson at the Large Hadron Collider (LHC) in Geneva, almost fifty years after it was first predicted as the source of the masses of elementary particles.

In the standard model, three of the four forces of nature -- the electromagnetic and the weak and strong nuclear forces -- arise from local gauge symmetry. The electromagnetic and weak forces are united in a single electroweak force, but its symmetry only holds it at the very high energies just now being reached at the LHC. It is spontaneously (accidentally) broken at lower energies. Gravity is still treated separately with Einstein's theory of general relativity, but it is also heavily based on symmetry principles.

While the standard model is a long way from Noether's original work, it confirms the general idea that what we call the laws of physics are simply logical requirements placed on our theories if we want them to be objective, that is, independent of the point of view of any particular observer. In my 2007 book The Comprehensible Cosmos. I called this principle point-of-view invariance and showed that virtually all of classical and quantum mechanics can be derived from it. The book subtitle is: Where Do the Laws of Physics Come From? The answer: They didn't come from anything. They are either the necessary requirement of symmetries that preserve point-of-view invariance or accidents that happen when these symmetries are broken.

Notes

The biographical material on Emmy Noether was taken from an article in the March 2013 issue of APSNews, a publication of the American Physical Society.

One of my mentors when I was a graduate student at UCLA in the early 1960s, theoretical physicist Nina Byers, has championed the place of Noether in physics and mathematical history. See Byers, 1996.

The development of particle physics from Democritus to the Higgs boson is discussed in my latest book, God and the Atom.